ABSTRACT
In this chapter, the authors consider the Cauchy problem for the system of isentropic gas dynamics in Eulerian coordinate. It provides theorems for the polytropic gas dynamics. The chapter gives the proof of the existence of viscosity solutions for the Cauchy problem. It constructs the weak entropy-entropy flux pairs for the polytropic case with the exponent. The chapter uses the compactness framework to study the river flow equations, a shallow-water model describing the vertical depth and mean velocity with bounded measurable initial data.
